Linear algebra

let A be nxn matrix.
Prove:
1. if A^2 = 0 so A columns are vectors in the solution space of the system Ax=0, and that the matrix deg is <= to n/2.
2. if deg(A^2)<deg(A), so to Ax=0 there is non trivial solution and that to A^2x=0 there is y that Ay =/ 0 (Ay is not equal to zero).
thanks ahead

sorry for the "bad" english, I'm don't know all the terms in english, so I try to translate(and It's uncorrect sometimes).
I wanted to know how I know if the columns space of A is in the solution space of Ax=0???
also, When you write nul(A) do you mean the solution space of Ax=0???
the second part is:
prove that if: rank(A^2) < rank(A) so to Ax=0 there is non-trivial solution!
and prove that: for A^2x=0 there is y that Ay=/0(non equal to zero)!
I checked!